Modified Dynamic Mode Decomposition (mDMD)

Abstract

Dynamical systems provide valuable insights into natural phenomena, classically modeled by mathematical equations. Recently, the analysis and characterization of dynamical systems is transitioning towards data-driven methods, such as Dynamic Mode Decomposition(DMD). The transition is propelled by the availability of large dynamic datasets from complex systems, often in the absence of explicit governing equations. The infinite dimensional linear Koopman operator offers a global linear approximation of nonlinear dynamics, providing a foundation for DMD-based approximations through finite-dimensional linear operators. In this thesis, we propose a nonhomogeneous extension to the existing DMD framework by incorporating a regularizer into the traditional homogeneous linear model. The proposed regularization enhances model flexibility, enabling the capture of external influences within dynamical systems. To implement this, we have solved an optimization problem, by minimizing the Frobenius norm of the finite dimensional Koopman operator with the added regularizer, thus finding optimal solutions to the nonhomogeneous linear model. Our proposed approach aims to achieve two primary data driven objectives: identifying coherent patterns within dynamic systems and extending the applicability of DMD to a wider range of nonlinear systems. Furthermore, we have numerically evaluated the influence of regularization within DMD, demonstrating that our modified DMD consistently outperforms the standard DMD methodology in a range of tested scenarios.